Discharge formula based on brink depth over sharp-crested weirs

The weir can be characterized as a hydraulic structure placed over open channels or rivers for several hydraulic uses. It is the most straightforward hydraulic construction for measuring, diverting, and regulating the flow (Khodier & Tullis 2022). In addition, it can be used for reasons such as flood passage (Borghei et al. 1999, Mahtabi & Arvanaghi 2018).

Weirs have been studied in recent decades and continue to attract the attention of several researchers due to their presence in numerous hydraulic constructions (Zhang et al. 2018). These investigations considered different shapes and kinds of weirs such as sharp-crested weirs (Li et al. 2021; Azimi & Salehi 2022; Azimi & Rajaratnam 2023; Li et al. 2023; Salehi et al. 2023), arched sharp-crested weir (Mirzaei & Sheibani 2021), compound sharp-crested (Zahiri et al. 2014), sharp and semicircular weirs crested with semicircular openings (Irzooki et al. 2014), sharp versus notch weir (Saadatnejadgharahassanlou et al. 2020a, 2020b; Nicosia et al. 2023), broad crested weir (Imanian et al. 2021), gear-shaped weir (Zhang et al. 2018; Alomari et al. 2023), circular-crested weir with a triangular plan form (Noori & Aaref 2017), piano key weir (Roushangar et al. 2021; Behroozi & Vaghefi 2022), labyrinth weir (Carrillo et al. 2020), and so many shapes and kinds. Despite the similarity of their definitions, their applications and hydraulic behaviour vary considerably (Borghei et al. 1999; Emiroglu & Kaya 2011).

Increasing the accuracy and efficiency of flow measurements is one technique to improve water management (Kassaye et al. 2022). When precise discharge measurements are necessary, sharp-crested weirs are commonly utilized to determine the flow discharge. It has both field and laboratory applications (Kumar et al. 2011). Recent studies have been conducted on various types and configurations of sharp-crested weirs for various applications. For instance, Wu & Rajaratnam (1996) experimentally investigated sharp-crested weir under full-width conditions. They separated the submerged flow into two sections: the impinging jet and the surface flow regime. They created a graphic to predict the various regimes. On the basis of their findings, Borghei et al. (1999) analysed 250 laboratory experiments and presented a discharge coefficient for sharp-crested side weir in a subcritical flow. Johnson (2000) conducted studies to determine the discharge coefficient for flow over flat-topped and sharp-crested weirs. He discovered that the coefficient may be used to forecast the discharge for weirs with flat-topped or sharp-crested edges. Afzalimehr & Bagheri (2010) estimated the discharge coefficient of rectangular sharp-crested weirs using the potential flow theory. They discovered that the proposed experimental equation for the discharge coefficient yielded reasonable results for a wide range of relative overflow depths. Bagheri & Heidarpour (2010) applied a free vortex theory to experimental data to estimate the discharge coefficient for sharp-crested weirs. The proposed theoretical analysis was found to correspond well with experimental observations. Aydin et al. (2011) empirically investigated the weir at various heights and widths. In the case of rectangular weirs, they discovered that the average flow velocity over the weir may be used instead of the discharge coefficient to calculate the discharge. Kumar et al. (2011) conducted studies to determine the discharge capacity of a sharp-crested triangular plan form weir under free-flow conditions. They discovered that the triangular plan form weirs were more effective than conventional weirs. During the circular sharp-crested weir investigation, Ghobadian & Meratifashi (2012) demonstrated that the disparity between the calculated and theoretical discharge for each stage of the weir increases with each step. A modification of the theoretical stage–discharge relation with the excellent agreement was obtained based on measured data. Mahtabi & Arvanaghi (2018) used the optimization method to establish the optimal discharge coefficient equation based on the statistical analysis. In addition, the accuracy of the numerical method used to simulate the flow over the sharp-crested weir was evaluated. When the ratio of water head over the weir crest to weir height is equal to or greater than 0.6, and the Reynolds number is similar to or more than 2000, the study found that the discharge coefficient is constant and equal to approximately 0.7. Al-Husseini et al. (2020) investigated experimentally and numerically inclined sharp-crested weirs and two types of vertical arch weirs (2020). Comparing these various types of weirs with standard sharp-crested weirs in terms of local scour depth revealed that the scour depth might be reduced thrice in the case of a 120° inclined sharp-crested weir.

Li & Han (2022) assessed the performance of several known discharge formulae of the characteristics of the flow discharge over a sharp-crested weir under free-flow conditions utilizing their experimental data and earlier data. The statistical analysis indicated that the weir velocity method proposed by Aydin et al. is suitable for representing discharges.

Brink depth or end depth (water depth at the end of the open channel) of freefall flow in open channels and hydraulic structures, which is usually located at the end of the channel of any hydraulic structures, has been utilized in the past (Rouse 1936) and is still being researched by numerous researchers for water discharge estimation (Abrari et al. 2019, Khosravinia et al. 2020, Afsharpour & Farhoudi 2021, Muhsin & Noori 2021). This depth can be used for flow metering as a single depth (Abrari et al. 2017, Zeidan et al. 2021). Ahmad & Azamathulla (2012) simulated the freefall from a circular channel and treated it as a flow over a sharp-crested, zero-height weir. The simulation anticipated the subcritical flow discharge based on end depth (brink depth). Using experimental data, the proposed equation was confirmed. Vatankhah (2013) established a theoretical relationship for forecasting the freefall water discharge in the trapezoidal open channel based on end depth (Brink depth) with an excellent agreement with experimental results. Khosravinia et al. (2020) studied the free fall flow discharge estimation in an open channel with a trapezoidal cross-section utilizing characteristics such as (h_b), channel bed and side slope, and bed width. This purpose was achieved by experimental data, and multiple computer techniques were developed based on numerous geometric circumstances. Direct and indirect estimations of freefall water of a semicircular cross-section open channel utilizing water depth at the brink were developed by Zeidan et al. (2021).

A simple formula with an accurate prediction of the flow discharge is essential from a practical and hydraulic viewpoint since it is straightforward to implement with high precision. Although numerous types and shapes of sharp-crested weirs have been extensively studied experimentally, numerically, and statistically, most proposed equations for calculating the water discharge over sharp-crested weirs are dependent on the water head at a particular distance upstream of the weir (h). However, measuring this water head at some field locations may be impossible due to the short length of the channel or the installation of a weir at the beginning or entry of the channel. In other instances, the water head upstream weir (h) may be affected by a nearby weir, the diversion channel, or other obstruction. In addition, the water head upstream weir (h) is affected by the channel bed slope, which is not considered in most published equations for predicting the water discharge over the sharp-crested weir. Therefore, the present work aims to experimentally investigate the sharp-crested weir to suggest a simple formula with high accuracy for estimating water discharge over the sharp-crested weirs based on brink depth (h_b) measured over the weir crest.

The volumetric calibrated weir, 10 cm in height, was installed temporarily at the start of each experiment and removed after water discharge was measured, and it was replaced by the research weir model.

This study examined five different heights (P) of a sharp, full-width weir: 10, 12, 14, 16, and 18 (five ratios of height to length of the weir (P/b): 0.33, 0.4, 0.47, 0.53, and 0.6). In the study, the bed slope was also taken into account. The bed slope was changed 10 times for the maximum and minimum weir heights, between 0.001 and 0.01. During all of the experiments, the freefall condition was maintained.

Steel rails supported a point gauge carriage along the top of the flume walls. The point gauge can move along the channel's length and width with this carriage's assistance. The water head over the weir was measured in each experiment using a point gauge (with a precision of up to 0.1 mm). Two places were used to gauge the water's head. These locations are at the brink of the weir crest, denoted by brink depth (hb), and upstream weir location, denoted by a head over the weir (h), as shown in Figure 2. The head over the weir is usually measured at a distance between three and four times the maximum value of the water head over the weir (to) (British Standards Institution BSI 3680-4A 1965), which is, in this study, less than 10 cm. Therefore, h was measured at a distance equal to about 35 cm upstream weir location.

This study considered 160 experiments of various models of rectangular sharp-crested weirs with changing bed slopes, water flows, and weir heights to develop an equation for discharge estimation using brink depth (h_b). The investigation also evaluated several (h/P) values ranging from 0.158 to 0.945.

As illustrated in the following equation, it is well known that the critical depth (y_c) of a uniform flow in a rectangular open channel has a straightforward relationship with unit discharge (q) (Chow 1959):

Experiments were conducted on a full-width standard sharp-crested weir with heights ranging from 10 to 18 cm, variable water discharge (between 1.54 and 17.25 L/s), and variable channel slopes (between 0.001 and 0.01). Various experiments yielded (h/P) values ranging from 0.158 to 0.94. For each experiment, the brink depth over the weir crest (h_b) and head over the weir (h) (upstream of the weir location) were measured to find a simple and accurate formula for predicting water discharge over the sharp-crested weir based on brink depth with various weir conditions, as mentioned previously.

Based on the experimental result and within its limitations, the following can be concluded:

• 1.

  Regarding the influence of the bed slope on the water head for a constant water discharge, h was inversely affected by the bed slope; this resulted in a reduction in water depth, which was observed upstream of the weir due to the increased bed slope. Increasing the slope of the bed will increase the flow velocity and then decrease the water head over the crest (h). Therefore, the accuracy of the prediction discharge equations, which predict water discharge based on the h value, is diminished since they do not take the bed slope into account. However, the brink depth (h_b) is not affected by changing the slope of the bed as it is located at zero distance from the location of the weir.

• 2.

  Concerning the development of a new formula to predict water discharge over the sharp-crested weir, a relationship between brink depth and critical depth was developed statically, within the experiment's limitations (h/P values between 0.158 and 0.9455), with an R² value of 0.9973. This relationship was then used to formulate a simple form formula based on the brink depth over the weir crest (h_e) (Equation (4)). This formula was examined by comparing it with other formulae such as the well known standard equation (Equation (5)) and the equation of Mahtabi & Arvanaghi (2018), Aydin et al. 2011, and Nicosia et al. 2023. The new formula has proven to be quite accurate in computing the discharge, with MAPE and RMSE values of 1.714% and 0.23, respectively.

Finally, it is necessary to evaluate the various types and shapes of the weir to examine the effect of bed slope on the water head and develop a new equation for calculating water discharge based on the brink depth, which is unaffected by bed slope.

The Department of Dams and Water Resources, College of Engineering, University of Mosul, Mosul, Iraq, supported this work. The authors would also like to thank all the staff members in the hydraulic laboratory at the University of Mosul.

All authors have read and agreed to the published version of the manuscript.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.